2020 ACJC Promo Q7

The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined as follows:

\(\mathrm{f}:x\mapsto 7{{\left( x-a \right)}^{2}}-1\), \(x\in \mathbb{R}\), \(x<3\)

\(\mathrm{g}:x\mapsto 1+b-{{e}^{-x}}\), \(x\ge 0\),

where \(a\) and \(b\) are real constants.

1. State the smallest value of \(a\) such that \({{\mathrm{f}}^{-1}}\) exists.

For the rest of this question, \(a=4\).

2. Explain if \({{\mathrm{f}}^{-1}}\) exists, and if it does, find \({{\mathrm{f}}^{-1}}\left( x \right)\) and state its domain.
3. Find the largest value of \(b\) such that the composite function \(\mathrm{fg}\) exists.
   For this value of \(b\), find \(\mathrm{fg}\left( x \right)\), state its domain, and find its range.